CAIE M2 (Mechanics 2) 2010 November

2
\includegraphics[max width=\textwidth, alt={}, center]{8425223a-1924-43ef-bd7c-e9b424fdc311-2_673_401_525_872} A bow consists of a uniform curved portion \(A B\) of mass 1.4 kg , and a uniform taut string of mass \(m \mathrm {~kg}\) which joins \(A\) and \(B\). The curved portion \(A B\) is an arc of a circle centre \(O\) and radius 0.8 m . Angle \(A O B\) is \(\frac { 2 } { 3 } \pi\) radians (see diagram). The centre of mass of the bow (including the string) is 0.65 m from \(O\). Calculate \(m\).

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\includegraphics[max width=\textwidth, alt={}, center]{8425223a-1924-43ef-bd7c-e9b424fdc311-2_279_905_1560_621} One end of a light inextensible string of length 0.2 m is attached to a fixed point \(A\) which is above a smooth horizontal surface. A particle \(P\) of mass 0.6 kg is attached to the other end of the string. \(P\) moves in a circle on the surface with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), with the string taut and making an angle of \(30 ^ { \circ }\) to the horizontal (see diagram).

1. Given that \(v = 1.5\), calculate the magnitude of the force that the surface exerts on \(P\).
2. Given instead that \(P\) moves with its greatest possible speed while remaining in contact with the surface, find \(v\).

4
\includegraphics[max width=\textwidth, alt={}, center]{8425223a-1924-43ef-bd7c-e9b424fdc311-3_560_894_258_628} A uniform beam \(A B\) has length 2 m and weight 70 N . The beam is hinged at \(A\) to a fixed point on a vertical wall, and is held in equilibrium by a light inextensible rope. One end of the rope is attached to the wall at a point 1.7 m vertically above the hinge. The other end of the rope is attached to the beam at a point 0.8 m from \(A\). The rope is at right angles to \(A B\). The beam carries a load of weight 220 N at \(B\) (see diagram).

1. Find the tension in the rope.
2. Find the direction of the force exerted on the beam at \(A\).

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\includegraphics[max width=\textwidth, alt={}, center]{8425223a-1924-43ef-bd7c-e9b424fdc311-4_433_841_255_653} A particle \(P\) is projected from a point \(O\) with initial speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(45 ^ { \circ }\) above the horizontal. \(P\) subsequently passes through the point \(A\) which is at an angle of elevation of \(30 ^ { \circ }\) from \(O\) (see diagram). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upward displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Write down expressions for \(x\) and \(y\) in terms of \(t\), and hence obtain the equation of the trajectory of \(P\).
2. Calculate the value of \(x\) when \(P\) is at \(A\).
3. Find the angle the trajectory makes with the horizontal when \(P\) is at \(A\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
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